Do your best to prove bona fide marriage. I think there is no such thing as "overkill" when it comes to proving good faith marriage. You definitely must file marriage certificate and child's birth certificate.
You are dealing with two sets of law here: family and immigration. Family law will help you deal with your kid's custody issue. Talk with a family law attorney in Houston Texas.
Immigration law: I want to advise you to talk with an immigration attorney about U visa to sort out your immigration status. You will need to report the domestic violence to a law enforcement officer.
Get help please. There are many non-profit organizations/shelters you can reach out. Good luck.